Optimal. Leaf size=43 \[ \frac {2 \sqrt {x^6+x}}{x^5+1}-2 \sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x^6+x}}\right ) \]
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Rubi [F] time = 1.75, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx &=\int \frac {x \left (-1+4 x^5\right )}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {1+x^5}\right ) \int \frac {\sqrt {x} \left (-1+4 x^5\right )}{\left (1+x^5\right )^{3/2} \left (a-x+a x^5\right )} \, dx}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+4 x^{10}\right )}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {4 x^2}{a \left (1+x^{10}\right )^{3/2}}+\frac {x^2 \left (-5 a+4 x^2\right )}{a \left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-5 a+4 x^2\right )}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^{10}\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}\\ &=\frac {8 x^2}{5 a \sqrt {x+x^6}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {5 a x^2}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )}+\frac {4 x^4}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}+\frac {\left (16 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{5 a \sqrt {x+x^6}}\\ &=\frac {8 x^2}{5 a \sqrt {x+x^6}}+\frac {16 x^2 \sqrt {1+x^5} \, _2F_1\left (\frac {3}{10},\frac {1}{2};\frac {13}{10};-x^5\right )}{15 a \sqrt {x+x^6}}-\frac {\left (10 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}\\ \end {align*}
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Mathematica [F] time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.70, size = 43, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x+x^6}}{1+x^5}-2 \sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x+x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 172, normalized size = 4.00 \begin {gather*} \left [\frac {{\left (x^{5} + 1\right )} \sqrt {a} \log \left (-\frac {a^{2} x^{10} + 2 \, a^{2} x^{5} + 6 \, a x^{6} - 4 \, {\left (a x^{5} + a + x\right )} \sqrt {x^{6} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{10} + 2 \, a^{2} x^{5} - 2 \, a x^{6} + a^{2} - 2 \, a x + x^{2}}\right ) + 4 \, \sqrt {x^{6} + x}}{2 \, {\left (x^{5} + 1\right )}}, \frac {{\left (x^{5} + 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{a x^{5} + a + x}\right ) + 2 \, \sqrt {x^{6} + x}}{x^{5} + 1}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {4 x^{6}-x}{\left (x^{5}+1\right ) \left (a \,x^{5}+a -x \right ) \sqrt {x^{6}+x}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, x^{6} - x}{{\left (a x^{5} + a - x\right )} \sqrt {x^{6} + x} {\left (x^{5} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {x-4\,x^6}{\left (x^5+1\right )\,\sqrt {x^6+x}\,\left (a\,x^5-x+a\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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